How to fix Non-Manifold elements?
As far as fixing nonmanifold issues are concerned, you don't need to worry about the UV unwrapping at all (except for possibly having to unwrap it again when you change your vertices around). 

The problem you have in the middle probably happened after you applied your mirror modifier, but it's an easy fix. Make sure just that section is selected, then scale it to 0 on the x axis. LEAVE IT SELECTED. Next you're going to go to your search bar (space in 2.79) and start typing in 'remove'. A few options should pop up, but what you want is remove doubles. You can also access it from the 'W' menu, but sometimes it just doesn't show up for me, so that's an alternative method.

Your other manifold problem looks like you might have internal faces. To check, go to face mode and see if you see any of those floating dots inside your model. If you do, move back into edge select and only select the edges on the inside (so you won't make holes in your model) and delete them.

I hope this helps!